YES Termination w.r.t. Q proof of Transformed_CSR_04_PALINDROME_nokinds-noand_Z.ari

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 QTRSRRRProof (⇔, 81 ms)
↳2 QTRS
↳3 QTRSRRRProof (⇔, 7 ms)
↳4 QTRS
↳5 RisEmptyProof (⇔, 0 ms)
↳6 YES

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
__(X, nil) → X
__(nil, X) → X
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U22(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNeList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt, P) → U72(isPal(activate(P)))
U72(tt) → tt
U81(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isPal(n__nil) → tt
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x1)) = x1   
POL(U21(x1, x2)) = x1 + x2   
POL(U22(x1)) = 1 + x1   
POL(U31(x1)) = 1 + x1   
POL(U41(x1, x2)) = x1 + x2   
POL(U42(x1)) = 1 + x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1)) = x1   
POL(U61(x1)) = x1   
POL(U71(x1, x2)) = 2·x1 + 2·x2   
POL(U72(x1)) = x1   
POL(U81(x1)) = 2 + 2·x1   
POL(__(x1, x2)) = 2·x1 + x2   
POL(a) = 2   
POL(activate(x1)) = x1   
POL(e) = 2   
POL(i) = 2   
POL(isList(x1)) = 1 + x1   
POL(isNeList(x1)) = 1 + x1   
POL(isNePal(x1)) = x1   
POL(isPal(x1)) = 2 + 2·x1   
POL(isQid(x1)) = x1   
POL(n____(x1, x2)) = 2·x1 + x2   
POL(n__a) = 2   
POL(n__e) = 2   
POL(n__i) = 2   
POL(n__nil) = 1   
POL(n__o) = 2   
POL(n__u) = 2   
POL(nil) = 1   
POL(o) = 2   
POL(tt) = 2   
POL(u) = 2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

__(X, nil) → X
__(nil, X) → X
U22(tt) → tt
U31(tt) → tt
U42(tt) → tt
U51(tt, V2) → U52(isList(activate(V2)))
U71(tt, P) → U72(isPal(activate(P)))
U81(tt) → tt
isPal(n__nil) → tt


(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

__(__(X, Y), Z) → __(X, __(Y, Z))
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U41(tt, V2) → U42(isNeList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U72(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Knuth-Bendix order [KBO] with precedence:
activate1 > u > o > i > isList1 > U212 > U221 > e > isNeList1 > U311 > isNePal1 > U611 > a > nil > nu > no > ni > ne > na > isPal1 > U811 > U712 > U512 > isQid1 > U421 > n2 > nnil > U721 > U521 > U412 > tt > U111 > _2

and weight map:

tt=16
n__nil=1
n__a=5
n__e=5
n__i=5
n__o=5
n__u=5
nil=2
a=6
e=6
i=6
o=6
u=6
U11_1=1
U22_1=2
isList_1=15
activate_1=1
U42_1=1
isNeList_1=13
U52_1=1
U61_1=1
U72_1=1
U31_1=1
isQid_1=11
isNePal_1=13
isPal_1=15
U81_1=1
___2=5
U21_2=2
U41_2=0
n_____2=4
U51_2=2
U71_2=10

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

__(__(X, Y), Z) → __(X, __(Y, Z))
U11(tt) → tt
U21(tt, V2) → U22(isList(activate(V2)))
U41(tt, V2) → U42(isNeList(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U72(tt) → tt
isList(V) → U11(isNeList(activate(V)))
isList(n__nil) → tt
isList(n____(V1, V2)) → U21(isList(activate(V1)), activate(V2))
isNeList(V) → U31(isQid(activate(V)))
isNeList(n____(V1, V2)) → U41(isList(activate(V1)), activate(V2))
isNeList(n____(V1, V2)) → U51(isNeList(activate(V1)), activate(V2))
isNePal(V) → U61(isQid(activate(V)))
isNePal(n____(I, __(P, I))) → U71(isQid(activate(I)), activate(P))
isPal(V) → U81(isNePal(activate(V)))
isQid(n__a) → tt
isQid(n__e) → tt
isQid(n__i) → tt
isQid(n__o) → tt
isQid(n__u) → tt
nil → n__nil
__(X1, X2) → n____(X1, X2)
a → n__a
e → n__e
i → n__i
o → n__o
u → n__u
activate(n__nil) → nil
activate(n____(X1, X2)) → __(X1, X2)
activate(n__a) → a
activate(n__e) → e
activate(n__i) → i
activate(n__o) → o
activate(n__u) → u
activate(X) → X


(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(6) YES